Kohler Pedestal Sink Install

I posted this in the remodeling forum, but probably should have posted here in the plumbing forum. Sorry for the double post.

Hello and thank you for your help

I am installing a Kohler Portrait pedestal sink and the back of the sink that mates up to the wall is very uneven. If I level the sink I have about a 1/4" inch gap between the wall and the top back of the sink. Should I level the sink or should I mount the sink so that it is flush to the wall?

I would level the sink and figure a way to deal with the gap. A couple of ideas come to mind which may or may not be applicable. Would it be possible to remove the wall covering (drywall?) and sister new studs that were plumb to the the existing studs then redo the wall? The other option would be to shim the gap and caulk it. The latter would be much easier, but wouldn't look as good as the former. There may be some other ways that I haven't thought of, so stay tuned.

The sink will be backing up to tile so I really can't sister 2x4's. I may be able to split the difference. I currently have a 1/8" shim between the pedestal and the bottom of the sink. I'll have to see if adding more will effect connecting the drain.